The name here comes from using web resources (referred to by url) as building blocks for a file system. I guess "Web for FS" rather than "Web as FS".
- NFS, particularly with the Solaris-originated concept of automounts over a /net mountpoint.
- Various virtual filesystems. Midnight Commander ("mc") offers several of these, including archive formats (tar, cpio, afio, rpm, deb) and remote (FTP, SSH).
- SMB/CFS/Samba
- Various FUSE filesystems, including again ssh, ftp, and others. These generally require specifying in advance specific mountpoints.
The notion of on-demand access to remote resources over protocols (e.g., http/https, or others), under filesystem dynamics, is interesting -- you can use any general tool, utility, or application for access, mediated through the filesystem by way of drivers, rather than a specific application (e.g., Web browser, FTP client, etc.)
There are numerous issues. In particular, applications tend not to respond well to remote resources disappearing, changing, or failing to return from change requests -- NFS's behaviour with nonresponsive remote hosts is ... notorious.
Consistency, availability, and partition resistance (CAP) are long-standing concerns, and there's no way to solve for all three. I'd add latency as another major consideration.
https://en.wikipedia.org/wiki/CAP_theorem
The general notion of managing and tracking changes locally, and pushing them to remote, has merits. I'm not aware of a "gitfs" ... though of course, one does exist, TIL: https://www.presslabs.com/code/gitfs/ The notion of using git (or other versioning system) as a mediator for remote/local revisioned access seems to have merits. Obviously not viable for very-high-variability systems, but adequate for many occasionally-modified resources.
I'm not sure if you're looking at using your WebFS itself as a publishing mechanism, though in general I think I'd recommend against doing this. For small-n peer-to-peer distribution that's probably workable, but for large-scale provisioning-and-request systems, relying on HTTP or other established transports is likely more sensible.
One area I've recognised as being particularly fraught is the whole notion of security and privacy. Providing unfiltered local access to remote resources which may change arbitrarily is a great way for allowing malware onto local systems -- your transport layer should probably implement some level of security and mounts deny direct execution of content. The fact that remote content could be copied to an executable mountpoint remains, and would make numerous attacks possible.
Similarly: access, update, write, and/or publishing actions all leak considerable information which could be of concern to specific users or organisations. Hash-based indexing (already addressed in this thread) being only one of several such vectors.
The point isn't whether or not these are protocols that are utilised on the naked Internet, but that they offer access to network services via filesystem semantics.
That is, rather than use a specific client or API to access remote content, or copying it locally as a separate step, you simply open a file in an existing application, or, within a program, using fopen() or equivalent operators. The networking is ... translucently ... handled in the background by the filesystem interface and/or driver(s).
The reasons SMB is not generally used or advised over the Internet are worth looking at, as this touches on many of the security / privacy concerns of any such service.